Question

Prove that every ideal in a principal ideal domain R (except Ritself) is contained in a maximal ideal.Exercise 10

Short answer

It is proved that every ideal in a principal ideal domain R is contained in a maximal ideal.

Step-by-step solution

Referring to Theorem 10.12, Lemma 10.9(1), and Exercise 10

Theorem 10.12

Let R be a principal ideal domain. Every non-zero, non-unit element of R is the product of irreducible elements, and this factorization is unique up to associates, i.e.,P₁P_2.......p_r= q₁q_2.........q_rif with each and irreducible then, r=s.

Lemma 10.9

Let a and b be elements of an integral domain R then,

(1) a⊂(b) if and only if b|a.

Exercise 10

An ideal of a PID is maximal if and only if p is irreducible.

Proving that every ideal in a principal ideal domain R is contained in a maximal ideal.

Let R be a PID, and (a) be an ideal in R.

Therefore, according to Theorem 10.12,a can be written as:

a=p₁p_2.......p_n

This implies,p_i|a.

Therefore, according to Lemma 10.9(1), a⊆ (p) .

Also, according to the result of Exercise 10, is maximal.

Hence, it is proved that every ideal in a principal ideal domain R is contained in a maximal ideal.